Mathematical Modelling FK (FRTN45)F2

Anders Rantzer

Department of Automatic Control, LTH

Modelling in three phases:

1. Problem structure◮ Formulate purpose, requirements for accuracy◮ Break up into subsystems — What is important?

2. Basic equations◮ Write down the relevant physical laws◮ Collect experimental data◮ Test hypotheses◮ Validate the model against fresh data

3. Model with desired features is formed◮ Put the model on suitable form.

(Computer simulation or pedagogical insight? )◮ Document and illustrate the model◮ Evaluate the model: Does it meet its purpose?

Implementation

Experiment Synthesis

Analysis

Matematical model

Idea/Purpose

specificationand requirement

Outline

Thursday lecture

◮ Course introduction◮ Ethics of modelling◮ Static models from data (black boxes)

Friday lecture

◮ Dynamic models from data (black boxes)◮ Models from physics (white boxes)◮ Mixed models (grey boxes)

Basic idea of system identification

S✲ ✲u y

Measure U and y. Figure out a model of S, consistent with measureddata.

Important aspects:

◮ We can only measure the u and y in discrete time points (sampling).Can be natural to use the discrete-time models.

◮ The system is affected by interference and measurement errors. Wemay need to signal models for dealing with this.

Example

A tank which attenuates flow variations in q1. Characterization of the tanksystem:

q1

q2

h

◮ Input: q1◮ Output: q2 and/or h◮ Internal variables / conditions: h

Step response

Step response for the tank 0 50 100 150 200 250 300 350 400−0.05

0

0.05

0.1

0.15

Can give idea of the dominant time constant, static reinforcement,character (overshoot or not)

Frequency response

For good signal-to-noise ratio, an estimate of G(iω) is obtained directlyfrom the amplitudes and phase positions of u, y

u(t) = A sinωty(t) = ApG(iω)p sin(ωt+ arg G(iω))

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How light affects pupil area Bode-diagram for pupil

Correlation analysis

Can we estimate the impulse response with other inputs?

◮ Impulse response formula in discrete time (T = 1, v = noise):

y(t) =∞∑

k=1�ku(t− k) + v(t)

◮ If v white noise with Ev2 = 1, then

Ryu(k) = Ey(t)u(t− k) = �k

◮ Covariance Ryu estimated by N data points with

RNyu(k) = 1

N

N∑

t=1y(t)u(t− k)

Example

Correlation analysis for 1s2+2s+1 (in- and out-put data)

Estimated and actual impulse responses Basic rules

Make experiments with conditions similar to the conditions in which themodel is to be used!

(Models from step response can be expected to work best on the stage.)

Save some data for model validation, i.e. check the model with data setdifferent from the one that generated the model!

Outline

Thursday lecture

◮ Course introduction◮ Ethics of modelling◮ Static models from data (black boxes)

Friday lecture

◮ Dynamic models from data (black boxes)◮ Models from physics (white boxes)◮ Mixed models (grey boxes)

Principles and analogies: Hydraulics

Example 1. A hydraulic system:

pa p1 p2 pb

Q1 Q2

Q3 Q4 Q5

Incompressible fluid. Pressures: pa, p1, p2, and p3.Volume flows: Q1, Q2, Q3, Q4, and Q5.

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Principles and analogies: Electrics

Example 2. An electrical system:

va v1 v2 vb

i1 i2

i3 i4 i5R3 R4 R5

C1 C2

Potentials va , vb , v1, and v2Currents i1 , i2, i3, i4, and i5

Principles and analogies: Heat

Example 3. A thermal system(heat transfer through a wall):

Värmekap. Värmekap.

Ta

T1 T2

Tbq3

q4 q5C1 C2

Two elements with thermal capacities C1 and C2 separated by insulatinglayers. Heat flows: q3 , q4 and q4Temperatures: Ta , Tb , T1 and T2

Principles and analogies: Mechanics

Exempel 4. A mechanical system:

Fa

F1 F2

Fb

v1 v2 v3

k1k2

d1 d2

m1 m2 m3

External forces: Fa and FbVelocities: v1, v2 and v3Spring constants: k1 and k2Damping constants: d1 and d2

Analogies

Analogies: hydraulic - electric - thermal - mechanicalTwo types of variables:

A. Flow Variables◮ volume flow◮ power flow◮ heat flow◮ speed

B. Intensity variables◮ pressure◮ voltage◮ temperature◮ force

For both of them addition rules hold.

Analogies (cont’d)

Intensity variations

C · ddt(intensity) = flow

C "capacitance":hydraulic: A/(ρ�)electrical: kapacitansheat: thermal capacitymechanical: inverse spring constantBalance equations!

(More complicated if the capacitance is not constant.)

Analogies (cont’d)

Lossesflow = φ(intensity)intensity = φ(flow)

Hydraulic: flow resistanceElectrics: resistanceHeat: thermal conductivityMechanics: friction

Often linear relationship in the electrical case - nonlinearly in the other(may be approximated by linear for small changes of variables)

More phenomena

Intensity variations

L · ddt(flow) = intensity

L "inductance"hydraucs: ρl/Aelectrics: inductansheat: –mechanics: massbalance equations!

(more complicated if the inductance is not constant.)

Energy flows

Can you make a general modeling theory based on flow and intensityvariables? Note the following.

pressure · flow = powervoltage difference · current = power

force · velocity = powertorque · angular velocity = powertemperature · heat flow = power · temperature

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Dimension analysis

Physical variables have dimensions. E.g.,

[density] = M L−3

[force] = M · LT2 = M LT−2

whereM = [mass], T = [time], L = [length]

Physical connections must be dimensionally “correct”.

Example: Bernoulli’s law

In Bernoulli’s law v =√

2�h you have

[v/√�h] = LT−1(LT−2 L)−0.5 = 1

v/√�h is an example of dimensionless quantity.

Dimensionless quantities and scaling

Some historical passanger ships:

◮ Kaiser Wilhelm the great, 1898, 22 knots, 200 m◮ Lusitania, 1909, 25 knots, 240 m◮ Rex, 1933, 27 knots, 269 m◮ Queen Mary, 1938, 29 knots, 311 m

Note that the ratio (velocity)2/(length) is almost constant

Which physical phenomenon can be thought to be the cause?

2 min problem

Find the relationship (except for a scaling by a dimensionless constant)between a pendulum period time and its mass, its length and theacceleration of gravity �, i.e.,

t = f (m, l, �)

A Graphical Modelica Model A submodel can be opened

Simple system in text format

Documentation is important:

Initialize at equilibrium!

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A predator-prey model

{

x = x(α − β y)y = y(δ x−γ )

Re-using old models

Inheritance:

Modfication:

Nonlinear differential-algebraic equations (DAE)

Differential-algebraic equations, DAE

F(z, z, u) = 0, y = H(z, u)

u: input, y: output, z: "internal variable"

Special case: state model

x = f (x, u), y = h(x, u)

u: input, y: output, x: state

Mathematics of general connection:

State models for two separate components:

φ1 = ω1 φ2 = ω2

J1ω1 = τ1 + τ2 J2ω2 = τ3 + τ4

Connection:

φ1 = φ2

τ2 = −τ3

The resulting model is not exactly a state model.

Linear differential-algebraic equations (DAE)

Ez = Fz+ Gu

If E were non-singular, one could write

z = E−1 Fz+ E−1Gu

which is a valid state model. If E is singular, variables have to beeliminated to get a state equation. Using a DAE solver is often better, sinceelimination can destroy sparsity.

Example:

1 0 0 00 J1 0 00 0 1 00 0 0 J20 0 0 00 0 0 0

φ1ω1φ2ω2

=

0 1 0 00 0 0 00 0 0 10 0 0 00 0 0 01 0 −1 0

φ1ω1φ2ω2

0 0 0 01 1 0 00 0 0 00 0 1 10 1 −1 00 0 0 0

τ1τ2τ3τ4

Outline

Thursday lecture

◮ Course introduction◮ Ethics of modelling◮ Static models from data (black boxes)

Friday lecture

◮ Dynamic models from data (black boxes)◮ Models from physics (white boxes)◮ Mixed models (grey boxes)

Prediction Error Methods

Find the unknown parameters θ by optimization:

minθqy(t,θ) − y(t)q

Here y(t) is the measured output at time t and y(t,θ) is the predictedoutput based on past measurements using a model with parameter valuesθ .

Prediction Error Method with Repeated Simulation

For a nonlinear grey-box model

0 = F(x, x, t,θ)y(t) = h(x, t,θ)

the unknown parameters θ could be determined by the prediction errormethod

minθqy(t,θ) − y(t)q

where the output prediction y(t,θ) is computed by simulation.

(Repeated simulation for different values of θ could however be verytime-consuming.)

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Population dynamics / Ecology

Variations in the number of lynx (solid) and hares (dashed) in Canada. Canyou predict the periodic variations?

Population dynamicsN1 number of lynx, N2 number of hares

ddt N1(t) = (λ1 −γ1)N1(t) +α1 N1(t)N2(t)ddt N2(t) = (λ2 −γ2)N1(t) −α2 N1(t)N2(t)

Simulation:

Mixing tanks in Skärblacka paper factory

A

B

A linear transfer function of three series-connected mixing tanks has theform 1

(sθ+1)3 .

To determine θ , radioactive lithium is added in A. Radioactivity was thenmeasured by B as a function of time.

Impulse response

In the lower picture, θ has been chosen to adapt to the impulse responseof 1(sθ+1)3

Grey Models — the best of both worlds

◮ White boxes: Physical laws provide some insight

◮ Black boxes: Statistics estimates complex relationships

◮ Gray boxes: Combine simplicity with insight

My Own Research: Dynamic Buffer Networks

◮ Producers, consumers and storages◮ Examples: water, power, traffic, data◮ Discrete/continuous, stochastic/deterministic◮ Multiple commodities, human interaction

Problem: Scalable and adaptive methods for control.

Example: Heating Networks

Grey box modelling!

Mathematical modelling — Lectures

Thursday lecture

◮ Course introduction◮ Ethics of modelling◮ Static models from data (black boxes)

Friday lecture

◮ Dynamic models from data (black boxes)◮ Models from physics (white boxes)◮ Mixed models (grey boxes)

Good luck with your projects!

I look forward to receiving your project plans.

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